Data Import and Cleaning

outliers = boxplot(df$price, plot=FALSE)$out
length(outliers)

[1] 1750

# df2<- df[which(df$price %in% outliers),]

Limitations of Dataset:

1. The “Belarus-Used-cars-catalog” data set is limited to only Belarus which in effect does not help Pattern 6 to make assumptions about used car markets in other countries.

2. There is no ‘electric’ car category as the data set is limited to gasoline and diesel.

3. There could have been more features found in the data set which Team Pattern 6 could have used for the Exploratory Data Analysis and get a more detailed analysis when comparing multiple features.

Normality tests

This section checks the normality of numerical variables based on the Q-Q plot, histogram, and normality tests. The most common method for normality test is called Shapiro-Wilk’s method, however, this test only works when the observation is less than 5000,and Belarus used car market data set is more extensive than this value, so a Kolmogorov-Smirnov (K-S) normality test will be used instead.

library(gridExtra)
plot1 = ggplot(cars_numerical, aes(sample = price)) + stat_qq(col="#00AFBB") + stat_qq_line() + labs(title = 'Q-Q plot of price') 
plot2 = ggplot(cars_numerical, aes(x = price)) + geom_histogram(fill = "#00AFBB", colour="white", bins=40) + labs(title = 'Histogram of price')

grid.arrange(plot1, plot2, ncol=2, nrow=1)

As it could be found in the quantile-quantile plot and the histogram,price are not normally distributed, if PatternSix wants to use the price as the dependent variable for a linear regression, it is necessary to transform it to a normal distribution after that.

plot3 = ggplot(cars_numerical, aes(sample = odometer_value)) + stat_qq(col="#00AFBB") + stat_qq_line() + labs(title = 'Q-Q plot of odometer_value')
plot4 = ggplot(cars_numerical, aes(x = odometer_value)) + geom_histogram(fill = "#00AFBB", colour="white", bins=40) + labs(title = 'Histogram of odometer_value')

plot5 = ggplot(cars_numerical, aes(sample = year_produced)) + stat_qq(col="#00AFBB") + stat_qq_line() + labs(title = 'Q-Q plot of year_produced')
plot6 = ggplot(cars_numerical, aes(x = year_produced)) + geom_histogram(fill = "#00AFBB", colour="white", bins=40) + labs(title = 'Histogram of year_produced')

grid.arrange(plot3, plot4, plot5, plot6, ncol=2, nrow=2)

plot7 = ggplot(cars_numerical, aes(sample = engine_capacity)) + stat_qq(col="#00AFBB") + stat_qq_line() + labs(title = 'Q-Q plot of engine_capacity')
plot8 = ggplot(cars_numerical, aes(x = engine_capacity)) + geom_histogram(fill = "#00AFBB", colour="white", bins=40) + labs(title = 'Histogram of engine_capacity')

plot9 = ggplot(cars_numerical, aes(sample = number_of_photos)) + stat_qq(col="#00AFBB") + stat_qq_line() + labs(title = 'Q-Q plot of number_of_photos')
plot10 = ggplot(cars_numerical, aes(x = number_of_photos)) + geom_histogram(fill = "#00AFBB", colour="white", bins=40) + labs(title = 'Histogram of number_of_photos')

grid.arrange(plot7, plot8, plot9, plot10, ncol=2, nrow=2)

The Q-Q plots and histograms also show evidence of non-normality. The odometer_value, engine_capacity and number_of_photos are right-skewed, while year_produced is left-skewed.

Now let’s apply Kolmogorov-Smirnov normality test into the data. The null hypothesis of this test is ‘sample distribution is normal’.

ks.test(df$price, 'pnorm', mean=mean(df$price), sd=sd(df$price))

    One-sample Kolmogorov-Smirnov test

data:  df$price
D = 0.2, p-value <2e-16
alternative hypothesis: two-sided

ks.test(df$odometer_value, 'pnorm', mean=mean(df$odometer_value), sd=sd(df$odometer_value))

    One-sample Kolmogorov-Smirnov test

data:  df$odometer_value
D = 0.06, p-value <2e-16
alternative hypothesis: two-sided

ks.test(df$year_produced, 'pnorm', mean=mean(df$year_produced), sd=sd(df$year_produced))

    One-sample Kolmogorov-Smirnov test

data:  df$year_produced
D = 0.06, p-value <2e-16
alternative hypothesis: two-sided

ks.test(df$engine_capacity, 'pnorm', mean=mean(df$engine_capacity), sd=sd(df$engine_capacity))

    One-sample Kolmogorov-Smirnov test

data:  df$engine_capacity
D = 0.2, p-value <2e-16
alternative hypothesis: two-sided

ks.test(df$number_of_photos, 'pnorm', mean=mean(df$number_of_photos), sd=sd(df$number_of_photos))

    One-sample Kolmogorov-Smirnov test

data:  df$number_of_photos
D = 0.1, p-value <2e-16
alternative hypothesis: two-sided

The p-value of all the numeric variables are < 2e-16 which is less than 0.05, therefore it could be concluded that the distributions of all our numeric variables are significantly different from normal distribution. They have the same results with Q-Q plots and histograms.

Our sample size for this data is 38521. Based on the central limit theorem, the rest analysis will be generated using the original data.

Correlation Plot

corrplot(cor(cars_numerical), method = 'number')

Figure 1 shows the correlation between the numerical features.

The team used a correlation plot for checking the correlation between continuous variables. Year of production was highly correlated with price with correlation coefficient(cc)=0.7. Odometer value had a negative correlation with year produced (cc=-0.49) and price (cc=-0.42). Engine capacity also had a positive correlation with price (cc=0.30).

library(ggplot2)

df %>% group_by(year_produced) %>% summarize(mean_price_per_year = mean(price, na.rm=TRUE)) %>% ggplot(aes(x=year_produced,y=mean_price_per_year)) +  geom_col(fill = "#00AFBB") + labs(title='Avg Price of Car per Year', x="year produced", y = "mean price per year") + theme(plot.title = element_text(hjust = 0.5))

Figure 2 shows the average price of the car for each year produced between 1940 and 2020. The team observed that there is a steady decrease in the price as the car gets older. However around 1990, it could be observed that the prices spike as cars before 1990 fall under the classic or vintage category.

The bar plot of the average price of the car in different years showed that the vintage cars produced around the year 1965 are pricier than the newer cars. And the price increased steadily after around 1985.

df %>% group_by(engine_capacity) %>% summarize(mean_price_per_capicity = mean(price, na.rm=TRUE)) %>% ggplot(aes(x=engine_capacity,y=mean_price_per_capicity)) +  geom_point(color = "#00AFBB") + labs(title='Avg Price of Car for engine capacity', x='Engine Capacity', y='Mean Price') + theme(plot.title = element_text(hjust = 0.5))

Figure 3 shows the average price of the car for each engine capacity. The team observed a positive linear trend between the mean price per engine capacity and the capacity

df %>% group_by(engine_capacity) %>% summarize(mean_price_per_capacity = mean(price, na.rm=TRUE)) ->df4
cor(df4)

                        engine_capacity mean_price_per_capacity
engine_capacity                   1.000                   0.583
mean_price_per_capacity           0.583                   1.000

#corrplot(cor(cars_numerical), method = 'number')

The observed correlation coefficient equals 0.6. However, in Figure 1 it was observed that the correlation coefficient between price and engine capacity was 0.3. This trend could be explained by the outliers which are found in higher engine capacity.

df %>% group_by(engine_capacity) %>% summarize(mean_price_per_capacity = mean(price, na.rm=TRUE)) ->df4
cor(df4)

                        engine_capacity mean_price_per_capacity
engine_capacity                   1.000                   0.583
mean_price_per_capacity           0.583                   1.000

#corrplot(cor(cars_numerical), method = 'number')

df %>% ggplot(aes(x=reorder(body_type,-engine_capacity),y=engine_capacity, fill=body_type))+geom_boxplot() + labs(x='Body Type', y='Engine Capicity')  + ggtitle('Body Type vs Engine Capicity ') + theme(plot.title = element_text(hjust = 0.5))

Figure 4 shows the mean engine capacity for different body type using a box-plot. From the initial analysis the team observed for each of the groups there is a difference in median.

T test

When there are two samples drawn from the same population and the goal is to test whether the mean of respective two samples are the same, it is wise to perform the student-t test, or t-test in short. The reason team PatternSix did not choose the Z-test is that the team did not know the population standard deviation. Thus using t-test, team used sample standard deviation (s) to estimate the population parameter (σ).

df %>% ggplot(aes(has_warranty, price, fill=has_warranty)) + geom_boxplot() + ggtitle('Has_Warranty vs Prices ') + theme(plot.title = element_text(hjust = 0.5))

summary(df$has_warranty)

False  True 
38072   449 

has = subset(df, has_warranty == "True")
hasnot = subset(df, has_warranty == "False")
# has = subset(df, has_warranty == 1)
# hasnot = subset(df, has_warranty == 0)
t.test(x = has$price, y = hasnot$price, conf.level = 0.99)

    Welch Two Sample t-test

data:  has$price and hasnot$price
t = 37, df = 452, p-value <2e-16
alternative hypothesis: true difference in means is not equal to 0
99 percent confidence interval:
 15907 18304
sample estimates:
mean of x mean of y 
    23543      6438 

df %>% ggplot(aes(engine_type, price,fill=engine_type)) + geom_boxplot()+ ggtitle('Engine_type vs Prices ') + theme(plot.title = element_text(hjust = 0.5))

summary(df$engine_type)

  diesel electric gasoline 
   12874        0    25647 

diesel = subset(df, subset = df$engine_type == "diesel")
gas = subset(df, subset = df$engine_type == "gasoline")
t.test(x = diesel$price, y = gas$price, conf.level = 0.99)

    Welch Two Sample t-test

data:  diesel$price and gas$price
t = 16, df = 24452, p-value <2e-16
alternative hypothesis: true difference in means is not equal to 0
99 percent confidence interval:
  981 1344
sample estimates:
mean of x mean of y 
     7411      6249 

\(Chi^2\) test

In the data set, not only do there are numerical variables,but there are also categorical variables. For categorical variables, data set does not fit the requirements for goodness of fit test but the data has to be tested for co-linearity between categorical variables for variable selection in model building. Test of Independence thus is performed.

contgcTbl1 = table(df$manufacturer_name, df$has_warranty)

(Xsq1 = chisq.test(contgcTbl1))

    Pearson's Chi-squared test

data:  contgcTbl1
X-squared = 10446, df = 54, p-value <2e-16

contgcTbl2 = table(df$manufacturer_name, df$body_type)

(Xsq2 = chisq.test(contgcTbl2))

    Pearson's Chi-squared test

data:  contgcTbl2
X-squared = 35332, df = 594, p-value <2e-16

contgcTbl3 = table(df$manufacturer_name, df$color)

(Xsq3 = chisq.test(contgcTbl3))

    Pearson's Chi-squared test

data:  contgcTbl3
X-squared = 6103, df = 594, p-value <2e-16

contgcTbl4 = table(df$color, df$transmission)

(Xsq4 = chisq.test(contgcTbl4))

    Pearson's Chi-squared test

data:  contgcTbl4
X-squared = 2381, df = 11, p-value <2e-16

contgcTbl5 = table(df$manufacturer_name, df$is_exchangeable)
(Xsq5 = chisq.test(contgcTbl5))

    Pearson's Chi-squared test

data:  contgcTbl5
X-squared = 436, df = 54, p-value <2e-16

The pairs that were chosen here are different manufacturers versus whether cars have warranties, different body types, different colors and whether cars are exchangeable, respectively. In addition, the test between different colors and whether the car is automatic or manual is also conducted. To make presenting results easier, these tests are assigned as 1, 2, 3, 4, 5 respectively. One thing to note here is that for the last test, to put which variable in row position or column position does not matter as a result of non casualty between them.

PatternSix’s null hypotheses are that all pairs are independent. Interestingly, wide range of results can be observed. For test 1, 2, 3, a warning that the chi-square test approximation might be incorrect pops up. The reason for that is to use the test of independence, sample size has to be large enough. General rule is that if expected frequencies for 20% of the categories are less than 5,it can’t be used to test independence. That is exactly what happened here. As a result, these test results can’t be used.

For test 4, between different manufacturers and whether cars are exchangeable, and for test 5, between different colors and whether the car is automatic or manual, the results are acceptable. Due to low p-values in both tests, the null hypothesis has been rejected, which means for test 4 and 5 testing pairs, they are not independent.

ANOVA

Due to the fact that there are numerous independent variables to test on, in order to improve efficiency, ANOVA was performed.

Same as above, a graph would give the observer an overview of relationships against prices.

df %>% group_by(color) %>% summarise(price_colorMean=mean(price)) %>% ggplot(aes(x=reorder(color,-price_colorMean),y=price_colorMean)) + geom_col(fill = "#00AFBB") + labs(x='Color',y='Price mean') + ggtitle('Color vs Prices ') + theme(plot.title = element_text(hjust = 0.5))

df %>% group_by(body_type) %>% summarise(body_price_mean = mean(price))%>% ggplot(aes(x = reorder(body_type, -body_price_mean),body_price_mean))+geom_col(fill = "#00AFBB") + labs(x='Body Type', y='Mean of price') + ggtitle('Body Type vs Price ') + theme(plot.title = element_text(hjust = 0.5))

df2 = df %>% group_by(manufacturer_name) %>% summarise(manuf_price_mean = mean(price)) %>% arrange(desc(manuf_price_mean)) 
df2 %>% slice(1:10) %>%  ggplot(aes(x = reorder(manufacturer_name, -manuf_price_mean),manuf_price_mean))+geom_col(fill = "#00AFBB") + labs(x='Manufacturer', y='Mean of price')  + ggtitle('Manufacturer vs Price ') + theme(plot.title = element_text(hjust = 0.5))

df_aov_1 = aov(price ~  color , df)
summary(df_aov_1)

               Df   Sum Sq  Mean Sq F value Pr(>F)    
color          11 1.70e+11 1.55e+10     420 <2e-16 ***
Residuals   38509 1.42e+12 3.69e+07                   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

df_aov_2 = aov(price ~  manufacturer_name , df)
summary(df_aov_2)

                     Df   Sum Sq  Mean Sq F value Pr(>F)    
manufacturer_name    54 2.94e+11 5.45e+09     162 <2e-16 ***
Residuals         38466 1.30e+12 3.37e+07                   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

df_aov_2 = aov(price ~  body_type , df)
summary(df_aov_2)

               Df   Sum Sq  Mean Sq F value Pr(>F)    
body_type      11 3.50e+11 3.18e+10     988 <2e-16 ***
Residuals   38509 1.24e+12 3.22e+07                   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Feautre selection

#Not working

# library("leaps")

# reg.best10 <- regsubsets(price~. , data = df, nvmax = 10, nbest = 1, method = "backward", really.big=T)  
# 
# # leaps::regsubsets() - Model selection by exhaustive (default) search, forward or backward stepwise, or sequential replacement
# #The plot will show the Adjust R^2 when using the variables across the bottom
# plot(reg.best10, scale = "adjr2", main = "Adjusted R^2")
# plot(reg.best10, scale = "r2", main = "R^2")
# # In the "leaps" package, we can use scale=c("bic","Cp","adjr2","r2")
# plot(reg.best10, scale = "bic", main = "BIC")
# plot(reg.best10, scale = "Cp", main = "Cp")
# summary(reg.best10)

# install.packages('party')
# library(party)
# cf1 <- cforest(price~. , data = df, control=cforest_unbiased(mtry=2,ntree=10))
# install.packages('relaimpo')
# library(relaimpo)
# lmMod <- lm(price~. , data = df, )  # fit lm() model
# relImportance <- calc.relimp(lmMod, type = "lmg", rela = TRUE) 

# library(caret)
# library(mlbench)
# 
# 
# test = df$price
# train = subset(df,select=-c(price,model_name,color,manufacturer_name))
# 
# 
# control <- rfeControl(functions=rfFuncs, method="cv", number=10)
# # run the RFE algorithm
# results <- rfe(train, test, rfeControl=control)
# # summarize the results
# print(results)
# # list the chosen features
# predictors(results)
# # plot the results
# plot(results, type=c("g", "o"))

Polynomial Terms

This has improved the fit but it will see whether changing some of the covariates can be helpful. The linearity assumption is checked by plots of residuals versus fitted values, plots of residuals versus explanatory variables, partial regression plots or gam plots. Here PatternSix apply the gam plot for our continuous variables to find the suitable transformation of the variables follow a linear model. The gam plots (one per explanatory variable) gives an idea which variables to transform: if the gam plot for a variable is straight, it suggests to leave that variable untransformed. If the plot for a particular variable is non-linear, the shape of the plot suggests the form of the transformation.

library(mgcv)
cal.gam <- gam(price_normal~s(odometer_value)+s(year_produced)+s(engine_capacity)+s(number_of_photos)+s(up_counter), data=df)

summary(fit1)

Call:
lm(formula = price ~ manufacturer_name + color + transmission + 
    odometer_value + engine_fuel + engine_capacity + body_type + 
    has_warranty + state + drivetrain + is_exchangeable + number_of_photos + 
    state + up_counter + year_produced, data = df)

Residuals:
   Min     1Q Median     3Q    Max 
-13519  -1626   -259   1100  36424 

Coefficients:
                                Estimate Std. Error t value Pr(>|t|)    
(Intercept)                    -8.33e+05   6.14e+03 -135.67  < 2e-16 ***
manufacturer_nameAlfa Romeo    -2.14e+03   4.67e+02   -4.60  4.3e-06 ***
manufacturer_nameAudi           3.96e+02   4.11e+02    0.96  0.33597    
manufacturer_nameBMW            8.22e+02   4.13e+02    1.99  0.04653 *  
manufacturer_nameBuick         -3.27e+03   6.30e+02   -5.19  2.1e-07 ***
manufacturer_nameCadillac      -2.71e+03   6.48e+02   -4.18  2.9e-05 ***
manufacturer_nameChery         -6.82e+03   5.94e+02  -11.49  < 2e-16 ***
manufacturer_nameChevrolet     -3.20e+03   4.35e+02   -7.35  2.0e-13 ***
manufacturer_nameChrysler      -2.65e+03   4.38e+02   -6.04  1.5e-09 ***
manufacturer_nameCitroen       -2.49e+03   4.16e+02   -5.98  2.2e-09 ***
manufacturer_nameDacia         -4.18e+03   5.91e+02   -7.07  1.5e-12 ***
manufacturer_nameDaewoo        -3.63e+03   4.64e+02   -7.83  5.1e-15 ***
manufacturer_nameDodge         -2.98e+03   4.49e+02   -6.63  3.4e-11 ***
manufacturer_nameFiat          -2.46e+03   4.24e+02   -5.79  6.9e-09 ***
manufacturer_nameFord          -1.96e+03   4.12e+02   -4.76  1.9e-06 ***
manufacturer_nameGeely         -5.96e+03   5.64e+02  -10.56  < 2e-16 ***
manufacturer_nameGreat Wall    -7.31e+03   6.83e+02  -10.69  < 2e-16 ***
manufacturer_nameHonda         -1.56e+03   4.23e+02   -3.68  0.00023 ***
manufacturer_nameHyundai       -2.44e+03   4.18e+02   -5.85  5.1e-09 ***
manufacturer_nameInfiniti      -1.03e+03   4.80e+02   -2.14  0.03214 *  
manufacturer_nameIveco          1.23e+03   5.04e+02    2.45  0.01448 *  
manufacturer_nameJaguar         5.21e+03   6.08e+02    8.57  < 2e-16 ***
manufacturer_nameJeep          -3.29e+03   5.15e+02   -6.38  1.8e-10 ***
manufacturer_nameKia           -2.68e+03   4.20e+02   -6.38  1.8e-10 ***
manufacturer_nameLADA          -5.25e+03   4.91e+02  -10.70  < 2e-16 ***
manufacturer_nameLancia        -2.36e+03   5.32e+02   -4.43  9.5e-06 ***
manufacturer_nameLand Rover     2.16e+01   4.73e+02    0.05  0.96361    
manufacturer_nameLexus          3.59e+03   4.65e+02    7.72  1.2e-14 ***
manufacturer_nameLifan         -5.76e+03   6.30e+02   -9.15  < 2e-16 ***
manufacturer_nameLincoln       -1.81e+03   7.20e+02   -2.51  0.01203 *  
manufacturer_nameMazda         -1.78e+03   4.16e+02   -4.29  1.8e-05 ***
manufacturer_nameMercedes-Benz  8.90e+02   4.14e+02    2.15  0.03130 *  
manufacturer_nameMini           7.79e+02   5.71e+02    1.36  0.17268    
manufacturer_nameMitsubishi    -2.34e+03   4.21e+02   -5.56  2.7e-08 ***
manufacturer_nameNissan        -2.27e+03   4.16e+02   -5.46  4.7e-08 ***
manufacturer_nameOpel          -2.02e+03   4.12e+02   -4.91  9.1e-07 ***
manufacturer_namePeugeot       -2.42e+03   4.14e+02   -5.84  5.3e-09 ***
manufacturer_namePontiac       -2.40e+03   6.49e+02   -3.69  0.00022 ***
manufacturer_namePorsche        2.38e+03   5.85e+02    4.07  4.7e-05 ***
manufacturer_nameRenault       -2.53e+03   4.13e+02   -6.13  9.1e-10 ***
manufacturer_nameRover         -2.10e+03   4.60e+02   -4.55  5.3e-06 ***
manufacturer_nameSaab          -1.88e+03   5.14e+02   -3.66  0.00026 ***
manufacturer_nameSeat          -1.54e+03   4.49e+02   -3.42  0.00063 ***
manufacturer_nameSkoda         -2.07e+03   4.26e+02   -4.86  1.2e-06 ***
manufacturer_nameSsangYong     -5.77e+03   5.50e+02  -10.48  < 2e-16 ***
manufacturer_nameSubaru        -3.05e+03   4.52e+02   -6.76  1.4e-11 ***
manufacturer_nameSuzuki        -3.53e+03   4.60e+02   -7.68  1.7e-14 ***
manufacturer_nameToyota         4.08e+01   4.16e+02    0.10  0.92186    
manufacturer_nameVolkswagen    -5.84e+02   4.10e+02   -1.42  0.15472    
manufacturer_nameVolvo         -7.26e+02   4.23e+02   -1.72  0.08623 .  
manufacturer_nameВАЗ           -1.78e+03   4.36e+02   -4.09  4.4e-05 ***
manufacturer_nameГАЗ            1.62e+03   4.73e+02    3.43  0.00060 ***
manufacturer_nameЗАЗ           -3.03e+03   6.52e+02   -4.64  3.5e-06 ***
manufacturer_nameМосквич        3.96e+03   6.10e+02    6.49  8.5e-11 ***
manufacturer_nameУАЗ           -6.97e+03   5.61e+02  -12.44  < 2e-16 ***
colorblue                      -3.15e+02   5.97e+01   -5.28  1.3e-07 ***
colorbrown                      8.72e+02   1.18e+02    7.41  1.3e-13 ***
colorgreen                     -2.73e+02   7.67e+01   -3.55  0.00038 ***
colorgrey                       7.67e+01   6.61e+01    1.16  0.24591    
colororange                     3.48e+02   2.49e+02    1.40  0.16191    
colorother                     -3.00e+02   7.49e+01   -4.00  6.2e-05 ***
colorred                        1.13e+02   7.47e+01    1.52  0.12892    
colorsilver                    -7.77e+02   5.60e+01  -13.89  < 2e-16 ***
colorviolet                    -2.68e+02   1.59e+02   -1.69  0.09085 .  
colorwhite                      3.74e+02   6.65e+01    5.62  2.0e-08 ***
coloryellow                    -6.49e+01   1.96e+02   -0.33  0.74074    
transmissionmechanical         -7.56e+02   4.79e+01  -15.78  < 2e-16 ***
odometer_value                 -5.82e-03   1.57e-04  -37.01  < 2e-16 ***
engine_fuelgas                 -1.19e+03   9.69e+01  -12.26  < 2e-16 ***
engine_fuelgasoline            -8.68e+02   4.27e+01  -20.32  < 2e-16 ***
engine_fuelhybrid-diesel        3.09e+03   2.33e+03    1.33  0.18452    
engine_fuelhybrid-petrol       -2.31e+02   2.29e+02   -1.01  0.31394    
engine_capacity                 8.27e+02   3.71e+01   22.32  < 2e-16 ***
body_typecoupe                 -2.09e+03   4.01e+02   -5.21  1.9e-07 ***
body_typehatchback             -3.93e+03   3.83e+02  -10.25  < 2e-16 ***
body_typeliftback              -2.79e+03   4.09e+02   -6.81  1.0e-11 ***
body_typelimousine             -2.16e+03   1.10e+03   -1.96  0.04988 *  
body_typeminibus               -1.03e+03   3.93e+02   -2.61  0.00906 ** 
body_typeminivan               -2.97e+03   3.86e+02   -7.69  1.5e-14 ***
body_typepickup                 1.13e+02   4.82e+02    0.23  0.81446    
body_typesedan                 -3.73e+03   3.82e+02   -9.77  < 2e-16 ***
body_typesuv                   -1.30e+03   3.87e+02   -3.35  0.00082 ***
body_typeuniversal             -3.86e+03   3.84e+02  -10.04  < 2e-16 ***
body_typevan                   -2.68e+03   4.02e+02   -6.66  2.7e-11 ***
has_warrantyTrue                1.42e+03   2.98e+02    4.75  2.0e-06 ***
statenew                        9.95e+03   3.49e+02   28.55  < 2e-16 ***
stateowned                      9.66e+02   1.73e+02    5.59  2.2e-08 ***
drivetrainfront                -2.16e+03   8.11e+01  -26.66  < 2e-16 ***
drivetrainrear                 -2.06e+03   9.54e+01  -21.58  < 2e-16 ***
is_exchangeableTrue            -1.97e+02   3.61e+01   -5.45  5.0e-08 ***
number_of_photos                7.40e+01   2.92e+00   25.37  < 2e-16 ***
up_counter                      2.48e+00   3.93e-01    6.32  2.7e-10 ***
year_produced                   4.22e+02   3.03e+00  139.00  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 3280 on 38428 degrees of freedom
Multiple R-squared:  0.74,  Adjusted R-squared:  0.739 
F-statistic: 1.19e+03 on 92 and 38428 DF,  p-value: <2e-16

par(mfrow=c(2,3))
plot(cal.gam)

These plots indicate that odometer_value and up_counter should remain untransformed (because the plot is relatively straight), but year_produced, engine_capacity and number_of_photos need transforming to generate a better result in the modeling. One possibility is to add a twice degree polynomial in year_produced, a sin transformation in engine_capacity and a twice degree polynomial in number_of_photos to the regression equation.

fit2 <- lm(price_normal ~ odometer_value+poly(year_produced, 2)+sin(engine_capacity)+poly(number_of_photos, 2)+up_counter
           +manufacturer_name+color+transmission+engine_fuel+body_type+has_warranty+state+drivetrain+is_exchangeable+state, data = df)
summary(fit2)

Call:
lm(formula = price_normal ~ odometer_value + poly(year_produced, 
    2) + sin(engine_capacity) + poly(number_of_photos, 2) + up_counter + 
    manufacturer_name + color + transmission + engine_fuel + 
    body_type + has_warranty + state + drivetrain + is_exchangeable + 
    state, data = df)

Residuals:
    Min      1Q  Median      3Q     Max 
-23.829  -1.277   0.094   1.353  24.017 

Coefficients:
                                Estimate Std. Error t value Pr(>|t|)    
(Intercept)                     2.87e+01   3.96e-01   72.64  < 2e-16 ***
odometer_value                 -1.11e-06   1.09e-07  -10.15  < 2e-16 ***
poly(year_produced, 2)1         9.26e+02   3.27e+00  283.27  < 2e-16 ***
poly(year_produced, 2)2         2.52e+02   2.79e+00   90.29  < 2e-16 ***
sin(engine_capacity)           -1.78e+00   3.97e-02  -44.75  < 2e-16 ***
poly(number_of_photos, 2)1      5.61e+01   2.35e+00   23.91  < 2e-16 ***
poly(number_of_photos, 2)2     -8.40e+00   2.22e+00   -3.78  0.00016 ***
up_counter                      2.14e-03   2.64e-04    8.11  5.0e-16 ***
manufacturer_nameAlfa Romeo    -2.20e+00   3.14e-01   -7.02  2.3e-12 ***
manufacturer_nameAudi           6.10e-01   2.76e-01    2.21  0.02726 *  
manufacturer_nameBMW            4.56e-01   2.77e-01    1.64  0.10001    
manufacturer_nameBuick         -3.89e+00   4.23e-01   -9.20  < 2e-16 ***
manufacturer_nameCadillac      -1.31e+00   4.34e-01   -3.03  0.00246 ** 
manufacturer_nameChery         -7.12e+00   3.99e-01  -17.84  < 2e-16 ***
manufacturer_nameChevrolet     -3.29e+00   2.92e-01  -11.25  < 2e-16 ***
manufacturer_nameChrysler      -2.31e+00   2.94e-01   -7.86  3.9e-15 ***
manufacturer_nameCitroen       -2.42e+00   2.80e-01   -8.66  < 2e-16 ***
manufacturer_nameDacia         -4.54e+00   3.97e-01  -11.43  < 2e-16 ***
manufacturer_nameDaewoo        -5.28e+00   3.11e-01  -16.94  < 2e-16 ***
manufacturer_nameDodge         -2.82e+00   3.02e-01   -9.35  < 2e-16 ***
manufacturer_nameFiat          -3.25e+00   2.85e-01  -11.43  < 2e-16 ***
manufacturer_nameFord          -2.88e+00   2.77e-01  -10.39  < 2e-16 ***
manufacturer_nameGeely         -7.10e+00   3.79e-01  -18.73  < 2e-16 ***
manufacturer_nameGreat Wall    -5.51e+00   4.59e-01  -12.00  < 2e-16 ***
manufacturer_nameHonda         -6.41e-01   2.84e-01   -2.26  0.02405 *  
manufacturer_nameHyundai       -2.48e+00   2.81e-01   -8.83  < 2e-16 ***
manufacturer_nameInfiniti      -9.42e-01   3.23e-01   -2.92  0.00350 ** 
manufacturer_nameIveco          1.22e-01   3.39e-01    0.36  0.71783    
manufacturer_nameJaguar         1.34e+00   4.08e-01    3.28  0.00105 ** 
manufacturer_nameJeep          -2.08e+00   3.46e-01   -6.02  1.8e-09 ***
manufacturer_nameKia           -2.75e+00   2.83e-01   -9.73  < 2e-16 ***
manufacturer_nameLADA          -6.42e+00   3.30e-01  -19.43  < 2e-16 ***
manufacturer_nameLancia        -2.39e+00   3.57e-01   -6.67  2.5e-11 ***
manufacturer_nameLand Rover    -7.36e-01   3.18e-01   -2.32  0.02046 *  
manufacturer_nameLexus          1.39e+00   3.12e-01    4.46  8.3e-06 ***
manufacturer_nameLifan         -6.74e+00   4.24e-01  -15.91  < 2e-16 ***
manufacturer_nameLincoln       -1.40e+00   4.83e-01   -2.90  0.00378 ** 
manufacturer_nameMazda         -1.99e+00   2.80e-01   -7.10  1.3e-12 ***
manufacturer_nameMercedes-Benz  6.75e-02   2.78e-01    0.24  0.80813    
manufacturer_nameMini           3.60e-01   3.84e-01    0.94  0.34840    
manufacturer_nameMitsubishi    -2.18e+00   2.83e-01   -7.73  1.1e-14 ***
manufacturer_nameNissan        -2.37e+00   2.79e-01   -8.47  < 2e-16 ***
manufacturer_nameOpel          -2.07e+00   2.77e-01   -7.47  8.0e-14 ***
manufacturer_namePeugeot       -1.93e+00   2.78e-01   -6.94  3.9e-12 ***
manufacturer_namePontiac       -1.29e+00   4.36e-01   -2.96  0.00303 ** 
manufacturer_namePorsche       -6.45e-02   3.93e-01   -0.16  0.86955    
manufacturer_nameRenault       -2.93e+00   2.77e-01  -10.57  < 2e-16 ***
manufacturer_nameRover         -2.74e+00   3.09e-01   -8.87  < 2e-16 ***
manufacturer_nameSaab          -9.74e-01   3.46e-01   -2.82  0.00485 ** 
manufacturer_nameSeat          -1.91e+00   3.02e-01   -6.34  2.3e-10 ***
manufacturer_nameSkoda         -1.98e+00   2.86e-01   -6.93  4.4e-12 ***
manufacturer_nameSsangYong     -3.61e+00   3.70e-01   -9.76  < 2e-16 ***
manufacturer_nameSubaru        -1.28e+00   3.04e-01   -4.22  2.4e-05 ***
manufacturer_nameSuzuki        -2.71e+00   3.09e-01   -8.78  < 2e-16 ***
manufacturer_nameToyota         3.70e-02   2.80e-01    0.13  0.89486    
manufacturer_nameVolkswagen    -6.98e-01   2.76e-01   -2.53  0.01128 *  
manufacturer_nameVolvo         -3.83e-01   2.85e-01   -1.35  0.17783    
manufacturer_nameВАЗ           -4.79e+00   2.93e-01  -16.35  < 2e-16 ***
manufacturer_nameГАЗ           -3.29e+00   3.22e-01  -10.21  < 2e-16 ***
manufacturer_nameЗАЗ           -6.95e+00   4.38e-01  -15.89  < 2e-16 ***
manufacturer_nameМосквич       -3.57e+00   4.13e-01   -8.63  < 2e-16 ***
manufacturer_nameУАЗ           -6.03e+00   3.77e-01  -16.01  < 2e-16 ***
colorblue                      -2.40e-01   4.01e-02   -5.98  2.3e-09 ***
colorbrown                     -1.67e-01   7.92e-02   -2.11  0.03474 *  
colorgreen                     -3.27e-01   5.16e-02   -6.34  2.4e-10 ***
colorgrey                      -1.42e-01   4.44e-02   -3.20  0.00136 ** 
colororange                    -1.92e-01   1.67e-01   -1.15  0.25068    
colorother                     -1.76e-01   5.03e-02   -3.50  0.00047 ***
colorred                       -5.55e-01   5.01e-02  -11.08  < 2e-16 ***
colorsilver                    -1.37e-01   3.77e-02   -3.64  0.00027 ***
colorviolet                    -3.90e-01   1.07e-01   -3.66  0.00026 ***
colorwhite                     -6.24e-01   4.47e-02  -13.95  < 2e-16 ***
coloryellow                    -2.77e-01   1.32e-01   -2.10  0.03539 *  
transmissionmechanical         -6.98e-01   3.20e-02  -21.81  < 2e-16 ***
engine_fuelgas                 -1.21e+00   6.51e-02  -18.58  < 2e-16 ***
engine_fuelgasoline            -1.14e+00   2.86e-02  -39.82  < 2e-16 ***
engine_fuelhybrid-diesel        3.27e+00   1.56e+00    2.09  0.03626 *  
engine_fuelhybrid-petrol       -1.13e+00   1.54e-01   -7.34  2.1e-13 ***
body_typecoupe                 -1.81e+00   2.70e-01   -6.73  1.7e-11 ***
body_typehatchback             -3.78e+00   2.57e-01  -14.70  < 2e-16 ***
body_typeliftback              -2.96e+00   2.75e-01  -10.77  < 2e-16 ***
body_typelimousine             -5.31e-01   7.39e-01   -0.72  0.47247    
body_typeminibus               -2.56e-01   2.64e-01   -0.97  0.33203    
body_typeminivan               -1.76e+00   2.59e-01   -6.78  1.2e-11 ***
body_typepickup                -4.50e-01   3.23e-01   -1.39  0.16386    
body_typesedan                 -3.48e+00   2.56e-01  -13.59  < 2e-16 ***
body_typesuv                   -1.51e+00   2.60e-01   -5.82  6.0e-09 ***
body_typeuniversal             -3.39e+00   2.58e-01  -13.15  < 2e-16 ***
body_typevan                   -1.62e+00   2.70e-01   -5.99  2.1e-09 ***
has_warrantyTrue               -7.61e-01   2.00e-01   -3.80  0.00014 ***
statenew                        5.81e+00   2.35e-01   24.76  < 2e-16 ***
stateowned                      4.76e+00   1.16e-01   41.00  < 2e-16 ***
drivetrainfront                -1.04e+00   5.45e-02  -19.08  < 2e-16 ***
drivetrainrear                 -5.55e-01   6.43e-02   -8.62  < 2e-16 ***
is_exchangeableTrue            -1.77e-01   2.42e-02   -7.31  2.7e-13 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2.2 on 38426 degrees of freedom
Multiple R-squared:  0.883, Adjusted R-squared:  0.883 
F-statistic: 3.1e+03 on 94 and 38426 DF,  p-value: <2e-16

plot(fit2,which=1)

Interaction Terms

Checking for interaction term between year_produced^2 *odometer_value. Removing color

fit3 <- lm(price_normal ~ odometer_value*poly(year_produced, 2)+sin(engine_capacity)+poly(number_of_photos, 2)+up_counter
           +manufacturer_name+color+transmission+engine_fuel+body_type+has_warranty+state+drivetrain+is_exchangeable+state, data = df)
summary(fit3)

Call:
lm(formula = price_normal ~ odometer_value * poly(year_produced, 
    2) + sin(engine_capacity) + poly(number_of_photos, 2) + up_counter + 
    manufacturer_name + color + transmission + engine_fuel + 
    body_type + has_warranty + state + drivetrain + is_exchangeable + 
    state, data = df)

Residuals:
   Min     1Q Median     3Q    Max 
-23.80  -1.27   0.08   1.34  23.27 

Coefficients:
                                        Estimate Std. Error t value Pr(>|t|)    
(Intercept)                             2.84e+01   3.93e-01   72.31  < 2e-16 ***
odometer_value                          1.29e-07   1.31e-07    0.99  0.32330    
poly(year_produced, 2)1                 8.66e+02   4.74e+00  182.67  < 2e-16 ***
poly(year_produced, 2)2                 2.88e+02   3.68e+00   78.25  < 2e-16 ***
sin(engine_capacity)                   -1.72e+00   3.95e-02  -43.44  < 2e-16 ***
poly(number_of_photos, 2)1              5.58e+01   2.33e+00   23.95  < 2e-16 ***
poly(number_of_photos, 2)2             -7.18e+00   2.21e+00   -3.25  0.00114 ** 
up_counter                              2.08e-03   2.62e-04    7.94  2.0e-15 ***
manufacturer_nameAlfa Romeo            -2.30e+00   3.11e-01   -7.38  1.6e-13 ***
manufacturer_nameAudi                   6.40e-01   2.74e-01    2.33  0.01962 *  
manufacturer_nameBMW                    4.18e-01   2.75e-01    1.52  0.12923    
manufacturer_nameBuick                 -3.76e+00   4.20e-01   -8.95  < 2e-16 ***
manufacturer_nameCadillac              -1.34e+00   4.31e-01   -3.11  0.00189 ** 
manufacturer_nameChery                 -7.09e+00   3.96e-01  -17.91  < 2e-16 ***
manufacturer_nameChevrolet             -3.25e+00   2.90e-01  -11.21  < 2e-16 ***
manufacturer_nameChrysler              -2.38e+00   2.92e-01   -8.16  3.4e-16 ***
manufacturer_nameCitroen               -2.50e+00   2.77e-01   -9.01  < 2e-16 ***
manufacturer_nameDacia                 -4.62e+00   3.94e-01  -11.74  < 2e-16 ***
manufacturer_nameDaewoo                -5.33e+00   3.09e-01  -17.25  < 2e-16 ***
manufacturer_nameDodge                 -2.90e+00   2.99e-01   -9.70  < 2e-16 ***
manufacturer_nameFiat                  -3.32e+00   2.82e-01  -11.75  < 2e-16 ***
manufacturer_nameFord                  -2.91e+00   2.75e-01  -10.57  < 2e-16 ***
manufacturer_nameGeely                 -7.02e+00   3.77e-01  -18.65  < 2e-16 ***
manufacturer_nameGreat Wall            -5.53e+00   4.56e-01  -12.13  < 2e-16 ***
manufacturer_nameHonda                 -7.18e-01   2.82e-01   -2.55  0.01091 *  
manufacturer_nameHyundai               -2.49e+00   2.79e-01   -8.95  < 2e-16 ***
manufacturer_nameInfiniti              -9.21e-01   3.20e-01   -2.88  0.00401 ** 
manufacturer_nameIveco                 -9.44e-02   3.36e-01   -0.28  0.77889    
manufacturer_nameJaguar                 1.40e+00   4.05e-01    3.46  0.00054 ***
manufacturer_nameJeep                  -2.11e+00   3.43e-01   -6.13  8.7e-10 ***
manufacturer_nameKia                   -2.76e+00   2.80e-01   -9.85  < 2e-16 ***
manufacturer_nameLADA                  -6.33e+00   3.28e-01  -19.26  < 2e-16 ***
manufacturer_nameLancia                -2.45e+00   3.55e-01   -6.92  4.7e-12 ***
manufacturer_nameLand Rover            -7.37e-01   3.15e-01   -2.34  0.01946 *  
manufacturer_nameLexus                  1.38e+00   3.10e-01    4.46  8.2e-06 ***
manufacturer_nameLifan                 -6.60e+00   4.21e-01  -15.70  < 2e-16 ***
manufacturer_nameLincoln               -1.41e+00   4.79e-01   -2.94  0.00333 ** 
manufacturer_nameMazda                 -2.03e+00   2.78e-01   -7.32  2.6e-13 ***
manufacturer_nameMercedes-Benz          6.73e-02   2.76e-01    0.24  0.80704    
manufacturer_nameMini                   3.67e-01   3.81e-01    0.96  0.33482    
manufacturer_nameMitsubishi            -2.24e+00   2.81e-01   -7.98  1.5e-15 ***
manufacturer_nameNissan                -2.40e+00   2.77e-01   -8.65  < 2e-16 ***
manufacturer_nameOpel                  -2.12e+00   2.75e-01   -7.72  1.2e-14 ***
manufacturer_namePeugeot               -2.00e+00   2.76e-01   -7.25  4.1e-13 ***
manufacturer_namePontiac               -1.38e+00   4.33e-01   -3.19  0.00143 ** 
manufacturer_namePorsche               -5.01e-02   3.90e-01   -0.13  0.89782    
manufacturer_nameRenault               -2.98e+00   2.75e-01  -10.81  < 2e-16 ***
manufacturer_nameRover                 -2.83e+00   3.07e-01   -9.21  < 2e-16 ***
manufacturer_nameSaab                  -1.06e+00   3.43e-01   -3.08  0.00205 ** 
manufacturer_nameSeat                  -1.98e+00   3.00e-01   -6.60  4.2e-11 ***
manufacturer_nameSkoda                 -2.04e+00   2.84e-01   -7.19  6.5e-13 ***
manufacturer_nameSsangYong             -3.65e+00   3.67e-01   -9.94  < 2e-16 ***
manufacturer_nameSubaru                -1.33e+00   3.02e-01   -4.41  1.0e-05 ***
manufacturer_nameSuzuki                -2.76e+00   3.07e-01   -9.00  < 2e-16 ***
manufacturer_nameToyota                -1.38e-02   2.78e-01   -0.05  0.96026    
manufacturer_nameVolkswagen            -7.11e-01   2.74e-01   -2.60  0.00934 ** 
manufacturer_nameVolvo                 -4.66e-01   2.82e-01   -1.65  0.09887 .  
manufacturer_nameВАЗ                   -4.91e+00   2.91e-01  -16.90  < 2e-16 ***
manufacturer_nameГАЗ                   -4.20e+00   3.22e-01  -13.03  < 2e-16 ***
manufacturer_nameЗАЗ                   -7.35e+00   4.35e-01  -16.90  < 2e-16 ***
manufacturer_nameМосквич               -4.83e+00   4.14e-01  -11.69  < 2e-16 ***
manufacturer_nameУАЗ                   -6.10e+00   3.74e-01  -16.32  < 2e-16 ***
colorblue                              -2.33e-01   3.99e-02   -5.85  4.9e-09 ***
colorbrown                             -1.04e-01   7.87e-02   -1.32  0.18647    
colorgreen                             -3.27e-01   5.14e-02   -6.37  1.9e-10 ***
colorgrey                              -1.19e-01   4.41e-02   -2.71  0.00681 ** 
colororange                            -1.10e-01   1.66e-01   -0.66  0.50607    
colorother                             -1.61e-01   4.99e-02   -3.22  0.00127 ** 
colorred                               -5.03e-01   4.98e-02  -10.10  < 2e-16 ***
colorsilver                            -1.59e-01   3.74e-02   -4.26  2.1e-05 ***
colorviolet                            -3.77e-01   1.06e-01   -3.57  0.00036 ***
colorwhite                             -5.73e-01   4.45e-02  -12.89  < 2e-16 ***
coloryellow                            -2.07e-01   1.31e-01   -1.58  0.11309    
transmissionmechanical                 -6.82e-01   3.18e-02  -21.49  < 2e-16 ***
engine_fuelgas                         -1.09e+00   6.49e-02  -16.87  < 2e-16 ***
engine_fuelgasoline                    -1.03e+00   2.89e-02  -35.83  < 2e-16 ***
engine_fuelhybrid-diesel                3.56e+00   1.55e+00    2.30  0.02156 *  
engine_fuelhybrid-petrol               -1.00e+00   1.53e-01   -6.57  5.2e-11 ***
body_typecoupe                         -1.81e+00   2.68e-01   -6.75  1.5e-11 ***
body_typehatchback                     -3.77e+00   2.55e-01  -14.76  < 2e-16 ***
body_typeliftback                      -2.94e+00   2.73e-01  -10.78  < 2e-16 ***
body_typelimousine                     -5.02e-01   7.34e-01   -0.68  0.49374    
body_typeminibus                       -3.08e-01   2.62e-01   -1.17  0.24015    
body_typeminivan                       -1.80e+00   2.57e-01   -7.00  2.7e-12 ***
body_typepickup                        -4.34e-01   3.21e-01   -1.35  0.17628    
body_typesedan                         -3.46e+00   2.54e-01  -13.60  < 2e-16 ***
body_typesuv                           -1.46e+00   2.58e-01   -5.64  1.7e-08 ***
body_typeuniversal                     -3.42e+00   2.56e-01  -13.38  < 2e-16 ***
body_typevan                           -1.59e+00   2.68e-01   -5.92  3.3e-09 ***
has_warrantyTrue                       -6.29e-01   1.99e-01   -3.15  0.00163 ** 
statenew                                6.03e+00   2.36e-01   25.56  < 2e-16 ***
stateowned                              4.73e+00   1.15e-01   41.11  < 2e-16 ***
drivetrainfront                        -1.02e+00   5.41e-02  -18.79  < 2e-16 ***
drivetrainrear                         -5.29e-01   6.38e-02   -8.28  < 2e-16 ***
is_exchangeableTrue                    -1.75e-01   2.41e-02   -7.29  3.2e-13 ***
odometer_value:poly(year_produced, 2)1  3.35e-04   2.10e-05   15.92  < 2e-16 ***
odometer_value:poly(year_produced, 2)2 -1.18e-04   1.77e-05   -6.67  2.6e-11 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2.19 on 38424 degrees of freedom
Multiple R-squared:  0.885, Adjusted R-squared:  0.885 
F-statistic: 3.09e+03 on 96 and 38424 DF,  p-value: <2e-16

plot(fit3,which=1)

Year_produced vs price

plot(df$year_produced, df$price, main="Year_produced vs price")

Year_produced vs price

plot(df$year_produced, df$price, main="Year_produced vs price", pch=19)

# Adding year_produced as polinomail of 2nd degree

fit2 <- lm(price ~ manufacturer_name+color+transmission+odometer_value+engine_fuel+engine_capacity+body_type+has_warranty+state+drivetrain+is_exchangeable+number_of_photos+state+up_counter+poly(year_produced,2), data = df)
summary(fit2)

Call:
lm(formula = price ~ manufacturer_name + color + transmission + 
    odometer_value + engine_fuel + engine_capacity + body_type + 
    has_warranty + state + drivetrain + is_exchangeable + number_of_photos + 
    state + up_counter + poly(year_produced, 2), data = df)

Residuals:
   Min     1Q Median     3Q    Max 
-23983  -1333    -33   1100  30808 

Coefficients:
                                Estimate Std. Error t value Pr(>|t|)    
(Intercept)                     9.92e+03   5.31e+02   18.68  < 2e-16 ***
manufacturer_nameAlfa Romeo    -1.44e+03   4.10e+02   -3.52  0.00043 ***
manufacturer_nameAudi          -1.13e+02   3.61e+02   -0.31  0.75354    
manufacturer_nameBMW            8.98e+02   3.62e+02    2.48  0.01321 *  
manufacturer_nameBuick         -4.41e+03   5.53e+02   -7.98  1.6e-15 ***
manufacturer_nameCadillac      -3.16e+03   5.68e+02   -5.56  2.7e-08 ***
manufacturer_nameChery         -7.18e+03   5.21e+02  -13.77  < 2e-16 ***
manufacturer_nameChevrolet     -3.94e+03   3.82e+02  -10.30  < 2e-16 ***
manufacturer_nameChrysler      -2.53e+03   3.84e+02   -6.57  5.0e-11 ***
manufacturer_nameCitroen       -2.38e+03   3.65e+02   -6.52  6.9e-11 ***
manufacturer_nameDacia         -4.72e+03   5.19e+02   -9.10  < 2e-16 ***
manufacturer_nameDaewoo        -2.96e+03   4.07e+02   -7.27  3.6e-13 ***
manufacturer_nameDodge         -2.83e+03   3.94e+02   -7.17  7.7e-13 ***
manufacturer_nameFiat          -2.05e+03   3.72e+02   -5.52  3.3e-08 ***
manufacturer_nameFord          -2.21e+03   3.62e+02   -6.12  9.7e-10 ***
manufacturer_nameGeely         -7.64e+03   4.95e+02  -15.42  < 2e-16 ***
manufacturer_nameGreat Wall    -7.15e+03   6.00e+02  -11.91  < 2e-16 ***
manufacturer_nameHonda         -1.20e+03   3.71e+02   -3.23  0.00123 ** 
manufacturer_nameHyundai       -2.52e+03   3.67e+02   -6.87  6.5e-12 ***
manufacturer_nameInfiniti      -1.23e+03   4.22e+02   -2.93  0.00342 ** 
manufacturer_nameIveco          7.23e+02   4.42e+02    1.63  0.10224    
manufacturer_nameJaguar         4.69e+03   5.34e+02    8.79  < 2e-16 ***
manufacturer_nameJeep          -3.09e+03   4.52e+02   -6.84  8.0e-12 ***
manufacturer_nameKia           -2.91e+03   3.69e+02   -7.90  2.9e-15 ***
manufacturer_nameLADA          -7.95e+03   4.32e+02  -18.42  < 2e-16 ***
manufacturer_nameLancia        -1.77e+03   4.67e+02   -3.78  0.00015 ***
manufacturer_nameLand Rover     7.80e+01   4.15e+02    0.19  0.85095    
manufacturer_nameLexus          3.25e+03   4.08e+02    7.97  1.6e-15 ***
manufacturer_nameLifan         -7.79e+03   5.53e+02  -14.08  < 2e-16 ***
manufacturer_nameLincoln       -2.14e+03   6.32e+02   -3.38  0.00071 ***
manufacturer_nameMazda         -1.66e+03   3.65e+02   -4.55  5.4e-06 ***
manufacturer_nameMercedes-Benz  5.33e+02   3.63e+02    1.47  0.14163    
manufacturer_nameMini           3.03e+02   5.01e+02    0.60  0.54609    
manufacturer_nameMitsubishi    -2.05e+03   3.69e+02   -5.55  2.8e-08 ***
manufacturer_nameNissan        -2.25e+03   3.65e+02   -6.15  7.7e-10 ***
manufacturer_nameOpel          -1.96e+03   3.62e+02   -5.42  6.1e-08 ***
manufacturer_namePeugeot       -2.19e+03   3.63e+02   -6.02  1.8e-09 ***
manufacturer_namePontiac       -2.18e+03   5.70e+02   -3.82  0.00013 ***
manufacturer_namePorsche        1.98e+03   5.13e+02    3.86  0.00011 ***
manufacturer_nameRenault       -2.71e+03   3.62e+02   -7.47  8.3e-14 ***
manufacturer_nameRover         -1.40e+03   4.04e+02   -3.46  0.00054 ***
manufacturer_nameSaab          -1.55e+03   4.51e+02   -3.42  0.00062 ***
manufacturer_nameSeat          -1.32e+03   3.94e+02   -3.34  0.00085 ***
manufacturer_nameSkoda         -2.71e+03   3.74e+02   -7.25  4.4e-13 ***
manufacturer_nameSsangYong     -5.30e+03   4.83e+02  -10.98  < 2e-16 ***
manufacturer_nameSubaru        -3.00e+03   3.97e+02   -7.56  4.2e-14 ***
manufacturer_nameSuzuki        -2.96e+03   4.04e+02   -7.33  2.3e-13 ***
manufacturer_nameToyota         1.51e+01   3.65e+02    0.04  0.96712    
manufacturer_nameVolkswagen    -1.03e+03   3.60e+02   -2.85  0.00435 ** 
manufacturer_nameVolvo         -6.85e+02   3.72e+02   -1.84  0.06532 .  
manufacturer_nameВАЗ           -2.73e+03   3.83e+02   -7.14  9.8e-13 ***
manufacturer_nameГАЗ           -6.28e+03   4.21e+02  -14.92  < 2e-16 ***
manufacturer_nameЗАЗ           -4.43e+03   5.73e+02   -7.74  9.9e-15 ***
manufacturer_nameМосквич       -4.14e+03   5.40e+02   -7.67  1.8e-14 ***
manufacturer_nameУАЗ           -7.56e+03   4.92e+02  -15.37  < 2e-16 ***
colorblue                       1.13e+01   5.24e+01    0.22  0.82957    
colorbrown                      1.75e+02   1.03e+02    1.69  0.09013 .  
colorgreen                      1.83e+02   6.74e+01    2.72  0.00663 ** 
colorgrey                      -2.69e+01   5.80e+01   -0.46  0.64271    
colororange                    -8.50e+01   2.18e+02   -0.39  0.69696    
colorother                     -1.20e+02   6.58e+01   -1.82  0.06813 .  
colorred                        1.46e+02   6.55e+01    2.23  0.02565 *  
colorsilver                    -3.82e+02   4.92e+01   -7.76  8.9e-15 ***
colorviolet                     2.68e+02   1.39e+02    1.92  0.05476 .  
colorwhite                      5.62e+00   5.85e+01    0.10  0.92349    
coloryellow                    -1.45e+02   1.72e+02   -0.84  0.39993    
transmissionmechanical         -5.18e+02   4.21e+01  -12.30  < 2e-16 ***
odometer_value                 -1.97e-03   1.43e-04  -13.81  < 2e-16 ***
engine_fuelgas                 -1.04e+03   8.50e+01  -12.29  < 2e-16 ***
engine_fuelgasoline            -8.52e+02   3.75e+01  -22.72  < 2e-16 ***
engine_fuelhybrid-diesel        2.07e+03   2.04e+03    1.01  0.31115    
engine_fuelhybrid-petrol       -6.36e+02   2.01e+02   -3.16  0.00158 ** 
engine_capacity                 1.41e+03   3.30e+01   42.82  < 2e-16 ***
body_typecoupe                 -1.99e+03   3.52e+02   -5.66  1.5e-08 ***
body_typehatchback             -3.60e+03   3.36e+02  -10.69  < 2e-16 ***
body_typeliftback              -2.92e+03   3.59e+02   -8.14  4.1e-16 ***
body_typelimousine             -1.56e+03   9.66e+02   -1.61  0.10638    
body_typeminibus               -8.05e+02   3.45e+02   -2.33  0.01970 *  
body_typeminivan               -2.79e+03   3.39e+02   -8.24  < 2e-16 ***
body_typepickup                -1.39e+03   4.23e+02   -3.28  0.00104 ** 
body_typesedan                 -3.83e+03   3.35e+02  -11.43  < 2e-16 ***
body_typesuv                   -1.89e+03   3.40e+02   -5.55  2.9e-08 ***
body_typeuniversal             -3.61e+03   3.37e+02  -10.69  < 2e-16 ***
body_typevan                   -2.56e+03   3.53e+02   -7.24  4.5e-13 ***
has_warrantyTrue               -6.43e+01   2.62e+02   -0.25  0.80600    
statenew                        8.03e+03   3.07e+02   26.19  < 2e-16 ***
stateowned                      1.32e+03   1.52e+02    8.69  < 2e-16 ***
drivetrainfront                -2.13e+03   7.12e+01  -29.95  < 2e-16 ***
drivetrainrear                 -2.35e+03   8.37e+01  -28.06  < 2e-16 ***
is_exchangeableTrue            -1.47e+02   3.17e+01   -4.63  3.6e-06 ***
number_of_photos                5.40e+01   2.57e+00   21.06  < 2e-16 ***
up_counter                      1.77e+00   3.45e-01    5.12  3.0e-07 ***
poly(year_produced, 2)1         7.53e+05   4.29e+03  175.49  < 2e-16 ***
poly(year_produced, 2)2         3.93e+05   3.67e+03  107.08  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2880 on 38427 degrees of freedom
Multiple R-squared:  0.799, Adjusted R-squared:  0.799 
F-statistic: 1.65e+03 on 93 and 38427 DF,  p-value: <2e-16

#poly(year_produced,2)

print(vif(fit2))

                        GVIF Df GVIF^(1/(2*Df))
manufacturer_name      24.43 54            1.03
color                   1.58 11            1.02
transmission            1.83  1            1.35
odometer_value          1.75  1            1.32
engine_fuel             1.64  4            1.06
engine_capacity         2.27  1            1.51
body_type               8.89 11            1.10
has_warranty            3.66  1            1.91
state                   3.76  2            1.39
drivetrain              6.19  2            1.58
is_exchangeable         1.06  1            1.03
number_of_photos        1.13  1            1.06
up_counter              1.03  1            1.02
poly(year_produced, 2)  3.48  2            1.37

We can see a significant increase in adjusted R-squared and lower Residual standard error Note: Annova to be performed on all models

plot(df$odometer_value, df$price, main="odometer_value vs price", pch=19)

Note: how to model 1/x^2 in lm

fit3 <- lm(price ~ manufacturer_name+color+transmission+engine_fuel+engine_capacity+body_type+has_warranty+state+drivetrain+is_exchangeable+number_of_photos+state+up_counter+poly(year_produced,2)+(1/odometer_value), data = df)
summary(fit2)

Call:
lm(formula = price ~ manufacturer_name + color + transmission + 
    odometer_value + engine_fuel + engine_capacity + body_type + 
    has_warranty + state + drivetrain + is_exchangeable + number_of_photos + 
    state + up_counter + poly(year_produced, 2), data = df)

Residuals:
   Min     1Q Median     3Q    Max 
-23983  -1333    -33   1100  30808 

Coefficients:
                                Estimate Std. Error t value Pr(>|t|)    
(Intercept)                     9.92e+03   5.31e+02   18.68  < 2e-16 ***
manufacturer_nameAlfa Romeo    -1.44e+03   4.10e+02   -3.52  0.00043 ***
manufacturer_nameAudi          -1.13e+02   3.61e+02   -0.31  0.75354    
manufacturer_nameBMW            8.98e+02   3.62e+02    2.48  0.01321 *  
manufacturer_nameBuick         -4.41e+03   5.53e+02   -7.98  1.6e-15 ***
manufacturer_nameCadillac      -3.16e+03   5.68e+02   -5.56  2.7e-08 ***
manufacturer_nameChery         -7.18e+03   5.21e+02  -13.77  < 2e-16 ***
manufacturer_nameChevrolet     -3.94e+03   3.82e+02  -10.30  < 2e-16 ***
manufacturer_nameChrysler      -2.53e+03   3.84e+02   -6.57  5.0e-11 ***
manufacturer_nameCitroen       -2.38e+03   3.65e+02   -6.52  6.9e-11 ***
manufacturer_nameDacia         -4.72e+03   5.19e+02   -9.10  < 2e-16 ***
manufacturer_nameDaewoo        -2.96e+03   4.07e+02   -7.27  3.6e-13 ***
manufacturer_nameDodge         -2.83e+03   3.94e+02   -7.17  7.7e-13 ***
manufacturer_nameFiat          -2.05e+03   3.72e+02   -5.52  3.3e-08 ***
manufacturer_nameFord          -2.21e+03   3.62e+02   -6.12  9.7e-10 ***
manufacturer_nameGeely         -7.64e+03   4.95e+02  -15.42  < 2e-16 ***
manufacturer_nameGreat Wall    -7.15e+03   6.00e+02  -11.91  < 2e-16 ***
manufacturer_nameHonda         -1.20e+03   3.71e+02   -3.23  0.00123 ** 
manufacturer_nameHyundai       -2.52e+03   3.67e+02   -6.87  6.5e-12 ***
manufacturer_nameInfiniti      -1.23e+03   4.22e+02   -2.93  0.00342 ** 
manufacturer_nameIveco          7.23e+02   4.42e+02    1.63  0.10224    
manufacturer_nameJaguar         4.69e+03   5.34e+02    8.79  < 2e-16 ***
manufacturer_nameJeep          -3.09e+03   4.52e+02   -6.84  8.0e-12 ***
manufacturer_nameKia           -2.91e+03   3.69e+02   -7.90  2.9e-15 ***
manufacturer_nameLADA          -7.95e+03   4.32e+02  -18.42  < 2e-16 ***
manufacturer_nameLancia        -1.77e+03   4.67e+02   -3.78  0.00015 ***
manufacturer_nameLand Rover     7.80e+01   4.15e+02    0.19  0.85095    
manufacturer_nameLexus          3.25e+03   4.08e+02    7.97  1.6e-15 ***
manufacturer_nameLifan         -7.79e+03   5.53e+02  -14.08  < 2e-16 ***
manufacturer_nameLincoln       -2.14e+03   6.32e+02   -3.38  0.00071 ***
manufacturer_nameMazda         -1.66e+03   3.65e+02   -4.55  5.4e-06 ***
manufacturer_nameMercedes-Benz  5.33e+02   3.63e+02    1.47  0.14163    
manufacturer_nameMini           3.03e+02   5.01e+02    0.60  0.54609    
manufacturer_nameMitsubishi    -2.05e+03   3.69e+02   -5.55  2.8e-08 ***
manufacturer_nameNissan        -2.25e+03   3.65e+02   -6.15  7.7e-10 ***
manufacturer_nameOpel          -1.96e+03   3.62e+02   -5.42  6.1e-08 ***
manufacturer_namePeugeot       -2.19e+03   3.63e+02   -6.02  1.8e-09 ***
manufacturer_namePontiac       -2.18e+03   5.70e+02   -3.82  0.00013 ***
manufacturer_namePorsche        1.98e+03   5.13e+02    3.86  0.00011 ***
manufacturer_nameRenault       -2.71e+03   3.62e+02   -7.47  8.3e-14 ***
manufacturer_nameRover         -1.40e+03   4.04e+02   -3.46  0.00054 ***
manufacturer_nameSaab          -1.55e+03   4.51e+02   -3.42  0.00062 ***
manufacturer_nameSeat          -1.32e+03   3.94e+02   -3.34  0.00085 ***
manufacturer_nameSkoda         -2.71e+03   3.74e+02   -7.25  4.4e-13 ***
manufacturer_nameSsangYong     -5.30e+03   4.83e+02  -10.98  < 2e-16 ***
manufacturer_nameSubaru        -3.00e+03   3.97e+02   -7.56  4.2e-14 ***
manufacturer_nameSuzuki        -2.96e+03   4.04e+02   -7.33  2.3e-13 ***
manufacturer_nameToyota         1.51e+01   3.65e+02    0.04  0.96712    
manufacturer_nameVolkswagen    -1.03e+03   3.60e+02   -2.85  0.00435 ** 
manufacturer_nameVolvo         -6.85e+02   3.72e+02   -1.84  0.06532 .  
manufacturer_nameВАЗ           -2.73e+03   3.83e+02   -7.14  9.8e-13 ***
manufacturer_nameГАЗ           -6.28e+03   4.21e+02  -14.92  < 2e-16 ***
manufacturer_nameЗАЗ           -4.43e+03   5.73e+02   -7.74  9.9e-15 ***
manufacturer_nameМосквич       -4.14e+03   5.40e+02   -7.67  1.8e-14 ***
manufacturer_nameУАЗ           -7.56e+03   4.92e+02  -15.37  < 2e-16 ***
colorblue                       1.13e+01   5.24e+01    0.22  0.82957    
colorbrown                      1.75e+02   1.03e+02    1.69  0.09013 .  
colorgreen                      1.83e+02   6.74e+01    2.72  0.00663 ** 
colorgrey                      -2.69e+01   5.80e+01   -0.46  0.64271    
colororange                    -8.50e+01   2.18e+02   -0.39  0.69696    
colorother                     -1.20e+02   6.58e+01   -1.82  0.06813 .  
colorred                        1.46e+02   6.55e+01    2.23  0.02565 *  
colorsilver                    -3.82e+02   4.92e+01   -7.76  8.9e-15 ***
colorviolet                     2.68e+02   1.39e+02    1.92  0.05476 .  
colorwhite                      5.62e+00   5.85e+01    0.10  0.92349    
coloryellow                    -1.45e+02   1.72e+02   -0.84  0.39993    
transmissionmechanical         -5.18e+02   4.21e+01  -12.30  < 2e-16 ***
odometer_value                 -1.97e-03   1.43e-04  -13.81  < 2e-16 ***
engine_fuelgas                 -1.04e+03   8.50e+01  -12.29  < 2e-16 ***
engine_fuelgasoline            -8.52e+02   3.75e+01  -22.72  < 2e-16 ***
engine_fuelhybrid-diesel        2.07e+03   2.04e+03    1.01  0.31115    
engine_fuelhybrid-petrol       -6.36e+02   2.01e+02   -3.16  0.00158 ** 
engine_capacity                 1.41e+03   3.30e+01   42.82  < 2e-16 ***
body_typecoupe                 -1.99e+03   3.52e+02   -5.66  1.5e-08 ***
body_typehatchback             -3.60e+03   3.36e+02  -10.69  < 2e-16 ***
body_typeliftback              -2.92e+03   3.59e+02   -8.14  4.1e-16 ***
body_typelimousine             -1.56e+03   9.66e+02   -1.61  0.10638    
body_typeminibus               -8.05e+02   3.45e+02   -2.33  0.01970 *  
body_typeminivan               -2.79e+03   3.39e+02   -8.24  < 2e-16 ***
body_typepickup                -1.39e+03   4.23e+02   -3.28  0.00104 ** 
body_typesedan                 -3.83e+03   3.35e+02  -11.43  < 2e-16 ***
body_typesuv                   -1.89e+03   3.40e+02   -5.55  2.9e-08 ***
body_typeuniversal             -3.61e+03   3.37e+02  -10.69  < 2e-16 ***
body_typevan                   -2.56e+03   3.53e+02   -7.24  4.5e-13 ***
has_warrantyTrue               -6.43e+01   2.62e+02   -0.25  0.80600    
statenew                        8.03e+03   3.07e+02   26.19  < 2e-16 ***
stateowned                      1.32e+03   1.52e+02    8.69  < 2e-16 ***
drivetrainfront                -2.13e+03   7.12e+01  -29.95  < 2e-16 ***
drivetrainrear                 -2.35e+03   8.37e+01  -28.06  < 2e-16 ***
is_exchangeableTrue            -1.47e+02   3.17e+01   -4.63  3.6e-06 ***
number_of_photos                5.40e+01   2.57e+00   21.06  < 2e-16 ***
up_counter                      1.77e+00   3.45e-01    5.12  3.0e-07 ***
poly(year_produced, 2)1         7.53e+05   4.29e+03  175.49  < 2e-16 ***
poly(year_produced, 2)2         3.93e+05   3.67e+03  107.08  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2880 on 38427 degrees of freedom
Multiple R-squared:  0.799, Adjusted R-squared:  0.799 
F-statistic: 1.65e+03 on 93 and 38427 DF,  p-value: <2e-16

#poly(year_produced,2)
library(car)
print(vif(fit1))

                   GVIF Df GVIF^(1/(2*Df))
manufacturer_name 20.43 54            1.03
color              1.53 11            1.02
transmission       1.83  1            1.35
odometer_value     1.64  1            1.28
engine_fuel        1.64  4            1.06
engine_capacity    2.21  1            1.49
body_type          8.71 11            1.10
has_warranty       3.65  1            1.91
state              3.73  2            1.39
drivetrain         6.17  2            1.58
is_exchangeable    1.06  1            1.03
number_of_photos   1.13  1            1.06
up_counter         1.03  1            1.02
year_produced      2.14  1            1.46

No effect of (1/odometer_value)

Checking for interaction term between year_produced^2 *odometer_value. Removing color

fit4 <- lm(price ~ manufacturer_name+transmission+engine_fuel+engine_capacity+body_type+has_warranty+state+drivetrain+is_exchangeable+number_of_photos+state+up_counter+poly(year_produced,2)*odometer_value, data = df)
summary(fit4)

Call:
lm(formula = price ~ manufacturer_name + transmission + engine_fuel + 
    engine_capacity + body_type + has_warranty + state + drivetrain + 
    is_exchangeable + number_of_photos + state + up_counter + 
    poly(year_produced, 2) * odometer_value, data = df)

Residuals:
   Min     1Q Median     3Q    Max 
-21253  -1321    -25   1120  30231 

Coefficients:
                                        Estimate Std. Error t value Pr(>|t|)    
(Intercept)                             1.01e+04   5.27e+02   19.17  < 2e-16 ***
manufacturer_nameAlfa Romeo            -1.39e+03   4.07e+02   -3.41  0.00064 ***
manufacturer_nameAudi                  -1.04e+02   3.59e+02   -0.29  0.77226    
manufacturer_nameBMW                    9.39e+02   3.61e+02    2.60  0.00922 ** 
manufacturer_nameBuick                 -4.80e+03   5.50e+02   -8.73  < 2e-16 ***
manufacturer_nameCadillac              -3.11e+03   5.65e+02   -5.50  3.9e-08 ***
manufacturer_nameChery                 -7.25e+03   5.18e+02  -13.98  < 2e-16 ***
manufacturer_nameChevrolet             -4.02e+03   3.80e+02  -10.58  < 2e-16 ***
manufacturer_nameChrysler              -2.47e+03   3.82e+02   -6.47  9.8e-11 ***
manufacturer_nameCitroen               -2.32e+03   3.63e+02   -6.38  1.8e-10 ***
manufacturer_nameDacia                 -4.53e+03   5.16e+02   -8.77  < 2e-16 ***
manufacturer_nameDaewoo                -2.88e+03   4.05e+02   -7.12  1.1e-12 ***
manufacturer_nameDodge                 -2.76e+03   3.92e+02   -7.04  2.0e-12 ***
manufacturer_nameFiat                  -2.00e+03   3.70e+02   -5.41  6.5e-08 ***
manufacturer_nameFord                  -2.20e+03   3.60e+02   -6.12  9.4e-10 ***
manufacturer_nameGeely                 -7.95e+03   4.93e+02  -16.12  < 2e-16 ***
manufacturer_nameGreat Wall            -7.14e+03   5.96e+02  -11.98  < 2e-16 ***
manufacturer_nameHonda                 -1.11e+03   3.69e+02   -3.01  0.00259 ** 
manufacturer_nameHyundai               -2.55e+03   3.65e+02   -6.98  3.0e-12 ***
manufacturer_nameInfiniti              -1.25e+03   4.19e+02   -2.99  0.00282 ** 
manufacturer_nameIveco                  9.94e+02   4.40e+02    2.26  0.02395 *  
manufacturer_nameJaguar                 4.64e+03   5.31e+02    8.74  < 2e-16 ***
manufacturer_nameJeep                  -3.07e+03   4.50e+02   -6.82  9.0e-12 ***
manufacturer_nameKia                   -2.95e+03   3.67e+02   -8.04  9.6e-16 ***
manufacturer_nameLADA                  -8.38e+03   4.30e+02  -19.50  < 2e-16 ***
manufacturer_nameLancia                -1.72e+03   4.64e+02   -3.69  0.00022 ***
manufacturer_nameLand Rover             1.34e+02   4.13e+02    0.32  0.74608    
manufacturer_nameLexus                  3.25e+03   4.06e+02    8.00  1.3e-15 ***
manufacturer_nameLifan                 -8.23e+03   5.51e+02  -14.94  < 2e-16 ***
manufacturer_nameLincoln               -2.16e+03   6.28e+02   -3.44  0.00058 ***
manufacturer_nameMazda                 -1.63e+03   3.63e+02   -4.48  7.6e-06 ***
manufacturer_nameMercedes-Benz          5.40e+02   3.61e+02    1.50  0.13461    
manufacturer_nameMini                   2.60e+02   4.98e+02    0.52  0.60103    
manufacturer_nameMitsubishi            -2.03e+03   3.67e+02   -5.52  3.4e-08 ***
manufacturer_nameNissan                -2.25e+03   3.63e+02   -6.19  5.9e-10 ***
manufacturer_nameOpel                  -1.91e+03   3.60e+02   -5.30  1.1e-07 ***
manufacturer_namePeugeot               -2.11e+03   3.61e+02   -5.84  5.4e-09 ***
manufacturer_namePontiac               -2.07e+03   5.67e+02   -3.65  0.00026 ***
manufacturer_namePorsche                2.06e+03   5.10e+02    4.04  5.4e-05 ***
manufacturer_nameRenault               -2.68e+03   3.60e+02   -7.45  9.5e-14 ***
manufacturer_nameRover                 -1.32e+03   4.02e+02   -3.28  0.00106 ** 
manufacturer_nameSaab                  -1.48e+03   4.49e+02   -3.30  0.00096 ***
manufacturer_nameSeat                  -1.26e+03   3.92e+02   -3.21  0.00134 ** 
manufacturer_nameSkoda                 -2.60e+03   3.72e+02   -6.98  3.0e-12 ***
manufacturer_nameSsangYong             -5.27e+03   4.80e+02  -10.96  < 2e-16 ***
manufacturer_nameSubaru                -2.95e+03   3.94e+02   -7.47  8.3e-14 ***
manufacturer_nameSuzuki                -2.90e+03   4.02e+02   -7.23  5.0e-13 ***
manufacturer_nameToyota                 5.77e+01   3.63e+02    0.16  0.87390    
manufacturer_nameVolkswagen            -9.74e+02   3.58e+02   -2.72  0.00650 ** 
manufacturer_nameVolvo                 -5.51e+02   3.70e+02   -1.49  0.13585    
manufacturer_nameВАЗ                   -2.47e+03   3.81e+02   -6.47  9.6e-11 ***
manufacturer_nameГАЗ                   -5.46e+03   4.22e+02  -12.94  < 2e-16 ***
manufacturer_nameЗАЗ                   -4.05e+03   5.70e+02   -7.12  1.1e-12 ***
manufacturer_nameМосквич               -3.01e+03   5.41e+02   -5.55  2.8e-08 ***
manufacturer_nameУАЗ                   -7.37e+03   4.89e+02  -15.08  < 2e-16 ***
transmissionmechanical                 -5.33e+02   4.19e+01  -12.73  < 2e-16 ***
engine_fuelgas                         -1.18e+03   8.48e+01  -13.92  < 2e-16 ***
engine_fuelgasoline                    -9.74e+02   3.78e+01  -25.81  < 2e-16 ***
engine_fuelhybrid-diesel                1.92e+03   2.03e+03    0.94  0.34550    
engine_fuelhybrid-petrol               -7.33e+02   2.00e+02   -3.66  0.00026 ***
engine_capacity                         1.46e+03   3.27e+01   44.62  < 2e-16 ***
body_typecoupe                         -1.97e+03   3.50e+02   -5.62  1.9e-08 ***
body_typehatchback                     -3.57e+03   3.35e+02  -10.66  < 2e-16 ***
body_typeliftback                      -2.96e+03   3.57e+02   -8.29  < 2e-16 ***
body_typelimousine                     -1.59e+03   9.60e+02   -1.65  0.09799 .  
body_typeminibus                       -7.01e+02   3.43e+02   -2.04  0.04106 *  
body_typeminivan                       -2.70e+03   3.37e+02   -8.03  1.0e-15 ***
body_typepickup                        -1.35e+03   4.21e+02   -3.20  0.00136 ** 
body_typesedan                         -3.84e+03   3.33e+02  -11.52  < 2e-16 ***
body_typesuv                           -1.94e+03   3.38e+02   -5.74  9.5e-09 ***
body_typeuniversal                     -3.55e+03   3.35e+02  -10.59  < 2e-16 ***
body_typevan                           -2.47e+03   3.51e+02   -7.05  1.8e-12 ***
has_warrantyTrue                       -4.59e+02   2.61e+02   -1.76  0.07849 .  
statenew                                7.12e+03   3.09e+02   23.04  < 2e-16 ***
stateowned                              1.36e+03   1.51e+02    9.04  < 2e-16 ***
drivetrainfront                        -2.19e+03   7.08e+01  -30.95  < 2e-16 ***
drivetrainrear                         -2.40e+03   8.33e+01  -28.77  < 2e-16 ***
is_exchangeableTrue                    -1.51e+02   3.15e+01   -4.78  1.8e-06 ***
number_of_photos                        5.48e+01   2.55e+00   21.50  < 2e-16 ***
up_counter                              1.80e+00   3.43e-01    5.26  1.4e-07 ***
poly(year_produced, 2)1                 8.54e+05   6.16e+03  138.71  < 2e-16 ***
poly(year_produced, 2)2                 3.95e+05   4.80e+03   82.22  < 2e-16 ***
odometer_value                         -4.16e-03   1.71e-04  -24.38  < 2e-16 ***
poly(year_produced, 2)1:odometer_value -6.21e-01   2.74e-02  -22.69  < 2e-16 ***
poly(year_produced, 2)2:odometer_value -1.89e-01   2.30e-02   -8.21  2.3e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2870 on 38436 degrees of freedom
Multiple R-squared:  0.801, Adjusted R-squared:  0.801 
F-statistic: 1.85e+03 on 84 and 38436 DF,  p-value: <2e-16

library(car)
print(vif(fit4))

                                       GVIF Df GVIF^(1/(2*Df))
manufacturer_name                     25.46 54            1.03
transmission                           1.83  1            1.35
engine_fuel                            1.68  4            1.07
engine_capacity                        2.26  1            1.50
body_type                              8.18 11            1.10
has_warranty                           3.68  1            1.92
state                                  3.90  2            1.41
drivetrain                             6.18  2            1.58
is_exchangeable                        1.06  1            1.03
number_of_photos                       1.13  1            1.06
up_counter                             1.03  1            1.02
poly(year_produced, 2)                12.69  2            1.89
odometer_value                         2.53  1            1.59
poly(year_produced, 2):odometer_value 11.15  2            1.83

We observe a increase in the Adjusted R-squared

Adding year_produced*number_of_photos

fit5 <- lm(price ~ manufacturer_name+transmission+engine_fuel+engine_capacity+body_type+has_warranty+state+drivetrain+is_exchangeable+state+up_counter+poly(year_produced,2)*odometer_value*number_of_photos, data = df)
summary(fit5)

Call:
lm(formula = price ~ manufacturer_name + transmission + engine_fuel + 
    engine_capacity + body_type + has_warranty + state + drivetrain + 
    is_exchangeable + state + up_counter + poly(year_produced, 
    2) * odometer_value * number_of_photos, data = df)

Residuals:
   Min     1Q Median     3Q    Max 
-18543  -1283     -8   1095  29201 

Coefficients:
                                                         Estimate Std. Error t value Pr(>|t|)    
(Intercept)                                              9.78e+03   5.22e+02   18.73  < 2e-16 ***
manufacturer_nameAlfa Romeo                             -1.40e+03   4.01e+02   -3.49  0.00048 ***
manufacturer_nameAudi                                   -7.37e+01   3.54e+02   -0.21  0.83495    
manufacturer_nameBMW                                     9.14e+02   3.55e+02    2.58  0.01002 *  
manufacturer_nameBuick                                  -5.31e+03   5.42e+02   -9.80  < 2e-16 ***
manufacturer_nameCadillac                               -3.11e+03   5.57e+02   -5.58  2.4e-08 ***
manufacturer_nameChery                                  -7.11e+03   5.10e+02  -13.92  < 2e-16 ***
manufacturer_nameChevrolet                              -3.95e+03   3.74e+02  -10.54  < 2e-16 ***
manufacturer_nameChrysler                               -2.49e+03   3.76e+02   -6.60  4.1e-11 ***
manufacturer_nameCitroen                                -2.32e+03   3.57e+02   -6.49  8.9e-11 ***
manufacturer_nameDacia                                  -4.42e+03   5.08e+02   -8.70  < 2e-16 ***
manufacturer_nameDaewoo                                 -2.85e+03   3.99e+02   -7.14  9.7e-13 ***
manufacturer_nameDodge                                  -2.78e+03   3.86e+02   -7.21  5.9e-13 ***
manufacturer_nameFiat                                   -2.06e+03   3.64e+02   -5.66  1.6e-08 ***
manufacturer_nameFord                                   -2.26e+03   3.54e+02   -6.38  1.8e-10 ***
manufacturer_nameGeely                                  -7.78e+03   4.85e+02  -16.03  < 2e-16 ***
manufacturer_nameGreat Wall                             -7.12e+03   5.87e+02  -12.13  < 2e-16 ***
manufacturer_nameHonda                                  -1.08e+03   3.64e+02   -2.97  0.00294 ** 
manufacturer_nameHyundai                                -2.44e+03   3.59e+02   -6.80  1.0e-11 ***
manufacturer_nameInfiniti                               -1.29e+03   4.13e+02   -3.14  0.00171 ** 
manufacturer_nameIveco                                   9.13e+02   4.33e+02    2.11  0.03504 *  
manufacturer_nameJaguar                                  4.29e+03   5.23e+02    8.21  2.3e-16 ***
manufacturer_nameJeep                                   -3.03e+03   4.43e+02   -6.84  8.1e-12 ***
manufacturer_nameKia                                    -2.90e+03   3.61e+02   -8.02  1.1e-15 ***
manufacturer_nameLADA                                   -8.00e+03   4.23e+02  -18.89  < 2e-16 ***
manufacturer_nameLancia                                 -1.70e+03   4.57e+02   -3.73  0.00019 ***
manufacturer_nameLand Rover                              2.17e+01   4.07e+02    0.05  0.95744    
manufacturer_nameLexus                                   3.19e+03   3.99e+02    8.00  1.3e-15 ***
manufacturer_nameLifan                                  -8.03e+03   5.42e+02  -14.80  < 2e-16 ***
manufacturer_nameLincoln                                -2.35e+03   6.19e+02   -3.79  0.00015 ***
manufacturer_nameMazda                                  -1.63e+03   3.58e+02   -4.55  5.3e-06 ***
manufacturer_nameMercedes-Benz                           5.14e+02   3.55e+02    1.45  0.14779    
manufacturer_nameMini                                   -1.57e+02   4.91e+02   -0.32  0.74929    
manufacturer_nameMitsubishi                             -2.05e+03   3.62e+02   -5.66  1.5e-08 ***
manufacturer_nameNissan                                 -2.20e+03   3.57e+02   -6.15  7.8e-10 ***
manufacturer_nameOpel                                   -1.89e+03   3.54e+02   -5.34  9.3e-08 ***
manufacturer_namePeugeot                                -2.09e+03   3.56e+02   -5.89  4.0e-09 ***
manufacturer_namePontiac                                -2.09e+03   5.58e+02   -3.74  0.00018 ***
manufacturer_namePorsche                                 2.04e+03   5.03e+02    4.06  4.9e-05 ***
manufacturer_nameRenault                                -2.65e+03   3.55e+02   -7.46  8.9e-14 ***
manufacturer_nameRover                                  -1.33e+03   3.96e+02   -3.36  0.00077 ***
manufacturer_nameSaab                                   -1.44e+03   4.42e+02   -3.27  0.00109 ** 
manufacturer_nameSeat                                   -1.30e+03   3.86e+02   -3.37  0.00077 ***
manufacturer_nameSkoda                                  -2.67e+03   3.67e+02   -7.28  3.3e-13 ***
manufacturer_nameSsangYong                              -5.21e+03   4.73e+02  -11.01  < 2e-16 ***
manufacturer_nameSubaru                                 -2.88e+03   3.88e+02   -7.41  1.3e-13 ***
manufacturer_nameSuzuki                                 -2.82e+03   3.95e+02   -7.14  9.5e-13 ***
manufacturer_nameToyota                                  7.22e+01   3.58e+02    0.20  0.84003    
manufacturer_nameVolkswagen                             -9.53e+02   3.53e+02   -2.70  0.00685 ** 
manufacturer_nameVolvo                                  -5.50e+02   3.64e+02   -1.51  0.13058    
manufacturer_nameВАЗ                                    -2.48e+03   3.75e+02   -6.61  3.8e-11 ***
manufacturer_nameГАЗ                                    -5.28e+03   4.16e+02  -12.70  < 2e-16 ***
manufacturer_nameЗАЗ                                    -4.10e+03   5.61e+02   -7.30  3.0e-13 ***
manufacturer_nameМосквич                                -2.52e+03   5.34e+02   -4.73  2.3e-06 ***
manufacturer_nameУАЗ                                    -7.33e+03   4.81e+02  -15.22  < 2e-16 ***
transmissionmechanical                                  -5.12e+02   4.12e+01  -12.40  < 2e-16 ***
engine_fuelgas                                          -1.16e+03   8.35e+01  -13.88  < 2e-16 ***
engine_fuelgasoline                                     -9.37e+02   3.72e+01  -25.17  < 2e-16 ***
engine_fuelhybrid-diesel                                 2.18e+03   2.00e+03    1.09  0.27519    
engine_fuelhybrid-petrol                                -7.09e+02   1.97e+02   -3.60  0.00032 ***
engine_capacity                                          1.50e+03   3.23e+01   46.63  < 2e-16 ***
body_typecoupe                                          -2.02e+03   3.45e+02   -5.86  4.7e-09 ***
body_typehatchback                                      -3.60e+03   3.30e+02  -10.91  < 2e-16 ***
body_typeliftback                                       -2.99e+03   3.52e+02   -8.49  < 2e-16 ***
body_typelimousine                                      -1.44e+03   9.46e+02   -1.52  0.12826    
body_typeminibus                                        -7.81e+02   3.38e+02   -2.31  0.02075 *  
body_typeminivan                                        -2.74e+03   3.32e+02   -8.26  < 2e-16 ***
body_typepickup                                         -1.29e+03   4.14e+02   -3.12  0.00178 ** 
body_typesedan                                          -3.87e+03   3.28e+02  -11.81  < 2e-16 ***
body_typesuv                                            -1.97e+03   3.33e+02   -5.92  3.2e-09 ***
body_typeuniversal                                      -3.59e+03   3.30e+02  -10.88  < 2e-16 ***
body_typevan                                            -2.60e+03   3.45e+02   -7.53  5.2e-14 ***
has_warrantyTrue                                        -8.33e+02   2.58e+02   -3.23  0.00124 ** 
statenew                                                 6.92e+03   3.06e+02   22.66  < 2e-16 ***
stateowned                                               1.50e+03   1.49e+02   10.07  < 2e-16 ***
drivetrainfront                                         -2.13e+03   6.98e+01  -30.56  < 2e-16 ***
drivetrainrear                                          -2.31e+03   8.21e+01  -28.10  < 2e-16 ***
is_exchangeableTrue                                     -2.24e+02   3.11e+01   -7.20  6.3e-13 ***
up_counter                                               1.95e+00   3.38e-01    5.77  8.0e-09 ***
poly(year_produced, 2)1                                  6.73e+05   9.78e+03   68.82  < 2e-16 ***
poly(year_produced, 2)2                                  2.91e+05   7.49e+03   38.85  < 2e-16 ***
odometer_value                                          -3.05e-03   2.91e-04  -10.51  < 2e-16 ***
number_of_photos                                         4.11e+01   6.68e+00    6.16  7.5e-10 ***
poly(year_produced, 2)1:odometer_value                  -2.13e-01   4.66e-02   -4.58  4.6e-06 ***
poly(year_produced, 2)2:odometer_value                  -9.76e-02   3.80e-02   -2.57  0.01030 *  
poly(year_produced, 2)1:number_of_photos                 1.73e+04   8.47e+02   20.44  < 2e-16 ***
poly(year_produced, 2)2:number_of_photos                 9.43e+03   6.74e+02   13.99  < 2e-16 ***
odometer_value:number_of_photos                         -5.79e-05   2.62e-05   -2.21  0.02734 *  
poly(year_produced, 2)1:odometer_value:number_of_photos -3.52e-02   3.99e-03   -8.80  < 2e-16 ***
poly(year_produced, 2)2:odometer_value:number_of_photos -3.20e-03   3.72e-03   -0.86  0.38875    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2820 on 38431 degrees of freedom
Multiple R-squared:  0.808, Adjusted R-squared:  0.807 
F-statistic: 1.81e+03 on 89 and 38431 DF,  p-value: <2e-16

library(car)
print(vif(fit5))

                                                        GVIF Df GVIF^(1/(2*Df))
manufacturer_name                                      26.40 54            1.03
transmission                                            1.83  1            1.35
engine_fuel                                             1.68  4            1.07
engine_capacity                                         2.27  1            1.51
body_type                                               8.22 11            1.10
has_warranty                                            3.71  1            1.93
state                                                   3.96  2            1.41
drivetrain                                              6.19  2            1.58
is_exchangeable                                         1.07  1            1.03
up_counter                                              1.03  1            1.02
poly(year_produced, 2)                                 80.26  2            2.99
odometer_value                                          7.55  1            2.75
number_of_photos                                        8.01  1            2.83
poly(year_produced, 2):odometer_value                  91.97  2            3.10
poly(year_produced, 2):number_of_photos                83.49  2            3.02
odometer_value:number_of_photos                        11.84  1            3.44
poly(year_produced, 2):odometer_value:number_of_photos 78.97  2            2.98

Very high VIFs for the new interaction terms

When the regression model passed either t test or F test, the only thing confirmed is the linear relationship between x and y is significant, or the regression model is valid. It does not gurantee the fit is good enough and we cannot rule out data unreliability due to other unknown factors, like outliers.

REGULARIZATION

library("ISLR")
df_reg = uzscale(df, append=0, "price_normal")
x=model.matrix(price_normal~.,df_reg)[,-1]
y=df_reg$price_normal

library("dplyr")
set.seed(1)
train = df_reg %>% sample_frac(0.75)
test = df_reg %>% setdiff(train)

x_train = model.matrix(price_normal~., train)[,-1]
x_test = model.matrix(price_normal~., test)[,-1]

y_train = train$price_normal %>% unlist()
y_test = test$price_normal %>% unlist()

# y_train = train %>% select(price) %>% unlist() # %>% as.numeric()
# y_test = test %>% select(price) %>% unlist() # %>% as.numeric()

library("glmnet")
grid=10^seq(10,-2,length=100)
ridge.mod=glmnet(x_train,y_train,alpha=0,grid=grid) 
plot(ridge.mod)

Using Grid search to find the optimal lambda value

# set.seed(1)
# ridge_cv=cv.glmnet(x_train,y_train,alpha=0, standardize = TRUE, nfolds = 10)  # Fit ridge regression model on training data
# 
# # Plot cross-validation results
# plot(ridge_cv)
#   
# # Best cross-validated lambda
# lambda_cv <- ridge_cv$lambda.min

Cross-validation is a statistical method for evaluating and comparing learning algorithms that divides data into two segments: one for learning or training a model and the other for validating it.

Cross validation involves the following steps:

1. Allocate a sample data set for study.
2. Use the rest of the dataset to train the model.
3. Use the test (validation) set’s reserve sample. This will assist you in determining how effective your model’s performance is. Proceed with the existing model if your model produces a positive result on validation data. It’s fantastic!

Performed prediction for the model, R-Squared test to check how model fits the set of observations and MSE test to check how close the estimates are close to the actual values.

# read the lambda value for cross-validation
lambda_cv <- 0.581
# Fit final model, get its sum of squared
# residuals and multiple R-squared
print("The value of lambda is for the lowest MSE is ")

[1] "The value of lambda is for the lowest MSE is "

print(lambda_cv)

[1] 0.581

model_cv <- glmnet(x_train,y_train, alpha = 0, lambda = lambda_cv,  standardize = TRUE)
y_hat_cv <- predict(model_cv, x_test)
ssr_cv <- t(y_test - y_hat_cv) %*% (y_test - y_hat_cv)

#R-sqaured
rsq_ridge_cv <- cor(y_test, y_hat_cv)^2
rsq_ridge_cv

        s0
[1,] 0.942

#MSE on test
mse0 <- mean((y_test - y_hat_cv) ^ 2)
sqrt(mse0)

[1] 1.56

print("R-sqaured")

[1] "R-sqaured"

print(rsq_ridge_cv)

        s0
[1,] 0.942

print("MSE on test")

[1] "MSE on test"

print(mse0)

[1] 2.43

grid=10^seq(10,-2,length=100)
lasso.mod=glmnet(x_train,y_train,alpha=1,grid=grid) 
plot(lasso.mod)

# # Center y, X will be standardized in the modelling function
# 
# # lambdas_to_try <- 10^seq(-3, 5, length.out = 100)
#   
# # Perform 10-fold cross-validation to select lambda 
# # Setting alpha = 1 implements lasso regression
# lasso_cv <- cv.glmnet(x_train, y_train, alpha = 1,  nfolds = 10)
#   
# # Plot cross-validation results
# plot(lasso_cv)
#   
# # Best cross-validated lambda
# lambda_cv <- lasso_cv$lambda.min
# print("Best lambda")
# print(lambda_cv)

# read the lambda value for cross-validation
lambda_cv <- 0.00374
# Fit final model, get its sum of squared 
model_cv_lasso <- glmnet(x, y, alpha = 1, lambda = lambda_cv, standardize = TRUE)
y_hat_cv <- predict(model_cv_lasso, x_test)
ssr_cv <- t(y_test - y_hat_cv) %*% (y_test - y_hat_cv)
rsq_lasso_cv <- cor(y_test, y_hat_cv)^2

plot(model_cv_lasso, xvar = "lambda")

print(ssr_cv)

      s0
s0 18907

print(rsq_lasso_cv)

        s0
[1,] 0.952